Consider a point $x \in \mathbb{R}^{3}$ and circles $C_{1}, C_{2} \subset \mathbb{R}^{3}$, where these three are all disjoint from each other. Then, consider the space $X=\mathbb{R}^{3}-\{x\}-C_{1}-C_{2}$. Compute the homology groups of $X$ using deformation retractions.
I've seen some solutions in the site for related problems that show that this would deformation retract to a wedge sum of some spaces, however I don't really see this. What would be a detailed explanation? (I'm kind of lost). Then, I think one would use Mayer-Vietoris sequences and cellular homology to compute the homology groups.
On a more general note, what would be a general "intuition" or method when deformation retracting a space to a simpler one for computations? I know the definitions, but not really a "method" or "algorithm" to use in order to simplify things (if that such thing exists in the first place)
By applying homotopies, we may assume that $x$ is the origin $O$ and $C_1$ and $C_2$ are latitudes of the unit sphere. Note that $\mathbb{R}^3 \setminus\{O\}$ is homotopy equivalent to $S^2$ via the map $y \mapsto \frac{y}{\|y\|}$. Removing the circles from the sphere leaves us with the disjoint union of three connected pieces: the top and bottom "caps" are contractible, and the "ring" in the middle is homotopy equivalent to $S^1$.